= Solution
Take an <Isothermal process>[isothermal] <Hydrostatic equilibrium>[hydrostatic] reference atmosphere, $\rho_0(z)=\rho_s e^{-z/H}$, and <Linearization>[linearize] about rest. The horizontal acceleration and pressure-gradient force retain the equatorial <Coriolis parameter> $f\simeq\beta y$, with $\beta=2\Omega/a$. The <traditional approximation> retains only its vertical component. Quadratic perturbation advection is neglected. The vertical acceleration is balanced by perturbation <buoyancy> and <Gradient>[pressure gradient] after removing the reference <Hydrostatic equilibrium>[hydrostatic balance].
The continuity constraint is the <anelastic approximation>, $\nabla\cdot(\rho_0\mathbf u)=0$, giving $u_x+v_y+w_z-w/H=0$. It retains stratification-driven expansion while filtering acoustic <density> fluctuations. It is therefore not the full compressible continuity equation. Adiabatic displacement in a stably stratified background gives $b_t+N^2w=0$, with $N$ the <Brunt–Väisälä frequency>; upward displacement develops restoring negative <buoyancy>.
\b[<Pressure> convention matters.] With the entropy-buoyancy equation just described, a consistent reduced-pressure form writes the <pressure> acceleration as $-\nabla\Phi$, where $\Phi=p'/\rho_0$. Its vertical term is $-\partial_z(p'/\rho_0)$, not simply $-\rho_0^{-1}p'_z$. The original source explicitly specifies variable $\rho_0(z)$ but prints the latter. Those differ by $p'/(\rho_0H)$, so the two cannot silently be equated. Parts <5/b> and <5/c> display the consequence and then solve the explicitly corrected reduced-pressure model that yields the requested wave equation. This is a mathematical inconsistency in the printed set, not missing source material.
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